Asymptotic Freedom We are familiar with the Coulomb interaction. Between two elementary charges the potential is 1 e2 V (r) 4 0 r It is convenient to write this in terms of the (dimensionless) fine structure constant, rather than the charge. 2 1 e c with V (r) 137 4 0c r However, we know from QED that the Coulomb force law (with being a constant) is actually an approximation for r not very small. At very small distances effective value of does increase slightly. Asymptotic Freedom The reason for the change in with e.g. Interaction distance has to do with the possible between two electrons photon exchanges that can occur. e At short distances the interaction is e mediated mostly by the exchange of single virtual photons. e e But at longer distances there is the possibility of more complicated e.g. e e (higher order) interactions. The interaction can include, not only photons, but also a “sea” of virtual e+ e electron positron pairs. This effect is known as “vacuum polarization”. e e Asymptotic Freedom The effect of the vacuum polarization is that the central charge is essentially screened, so that at large distances a lower charge is seen by another charge. i.e. the potential has a form like (r)c V (r) Where r 1 (r) 137 The usual value of r This is a very small effect. But at very small distances the “bare” or “unscreened” charge is seen. Asymptotic Freedom In particle physics it is more common to write the coupling constant in terms of the momentum transfer, q, that occurs from one charge to another. This is more intuitive when considering the scattering of one charge from another. ( 2 ) Then from QED: (q2 ) ( 2 ) q2 1 ln 2 3 2 where ( ) is the value of (q2 ) at some value of q 1 Long distances correspond to q 0 so (0) 137 Using 2 is just a way of parameterising how changes with q2. We cannot use = 0 since the equation blows up. q2 2 ln ln(1) 0 (q ) (0) But with q 0 2 and Asymptotic Freedom The change in the coupling constant is small enough to be ignored in all but the highest energy interactions. e.g. at a momentum q 10 GeV/c (q2 ) 1/135 But it is necessary for high precision calculations. e.g. The electron’s magnetic moment is related to its spin by S g with g = 2.00232 rather than g = 2 B This is effectively because the electron charge that is important in this relation is the bare (unscreened) charge not the effective charge at large distances. Asymptotic Freedom The screening effect is even more important in the strong interaction – quantum chromodynamics (QCD). The interaction between two quarks can include a single gluon exchange plus many additional diagrams. i.e. beyond we can have… q q q q q q q q q etc… q Asymptotic Freedom Because the gluons themselves carry colour charge, they do not “screen” the colour charge between quarks, but effectively add to it. The full QCD calculation including the vacuum polarization diagrams yields. c V (r) s r With the strong interaction coupling constant 2 2 s ( ) 2 2 s (q ) for q (11n 2 f ) ( 2 ) q2 1 s ln 2 3 4 This is just like the QED result with some reference momentum transfer. In this general result: n = number of colours and f = number of quark flavors in the theory. So if 11n > 2f the theory results in anti-screening. Asymptotic Freedom In the standard model n = 3 and f = 6, (11n - 2f )/3 = 7 > 0, therefore, because of the plus sign in the denominator 2 s decreases with increasing q . i.e. s increases with increasing distance. Therefore, unlike QED, we cannot2 choose 0 as our 2 s ( ) 2 2 references (q ) value since that is where s getsfor q larger than 1. (11n 2 f ) ( 2 ) q2 If we define 2 2 s 12 ln1 ln ln 2 3 4 2 (11n 2 f)s ( ) we can write the running coupling constant in terms of the single parameter . 2 12 2 2 s (q ) for q q2 (11n 2 f )ln 2 Asymptotic Freedom The value of must be found from fitting data. Unfortunately it is not very precisely known. 200 MeV/c with uncertainty 100 MeV/c 500 MeV/c 2 1.8 Thus we see that 2 2 12 (q ) 1.6 s (q ) s q2 1.4 (11n 2 f )ln at high 2 momentum 1.2 1 transfers (short 0.8 distances) the 0.6 0.4 interaction 0.2 0 between quarks is 0 5 10 15 20 relatively weak. q (GeV/c) This is the meaning of asymptotic freedom. Asymptotic Freedom The fact that s is relatively small at high energies means that s can be used as a perturbation parameter in calculations of high energy processes using QCD. i.e. fewer high order Feynman diagrams need to be included at high energies. Thus high energy process may be analysed quantitatively using QCD, but low energy process still cannot reliably be calculated. e.g. the nucleon-nucleon interaction cannot be calculated using QCD. e.g. the leading order QCD correction to the process e e hadrons (which includes the radiation of a gluon giving rise to 3 jet events) yields (ee hadrons) (E 2 / c2 ) R(E) 3 Q2 1 s f (e e ) f which provides better agreement with data. Asymptotic Freedom Feynman provided a different way of interpreting the result of asymptotic freedom. Consider the process of an electron scattering off a proton. At low momentum transfers e e the wavelength of the virtual photon is long and so the proton is seen as approximately a point charge and so the quark substructure is not apparent. p p Asymptotic Freedom As the momentum transfer increases, the virtual e photon wavelength e becomes similar to the distance between quarks. The energy is shared between the quarks. q q The interaction energy between the quarks is not q very high. So an analysis of the system is very complicated. Asymptotic Freedom At high momentum transfers the wavelength of the virtual photon is e e small enough that it interacts with just one quark. It then has a high energy relative to the other q quarks. The coupling to the other q q quarks is now not so important. Therefore the one quark can be considered to act as if it is “free”. .